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A Simple Harmonic Oscillator in a Vacuum

To demonstrate the method used in this paper, we first investigate the energetics of an oscillator which executes simple harmonic motion around its mean position without external interferences:

$$h(t) = a \cos ( \omega t + \phi ) ,$$ (3)

where h(t) is the position of the oscillator as a function of time t, a is the amplitude of oscillation, $\omega$ is the frequency of vibration, and $\phi$ is the phase. This models the motion of a single molecular machine ``pin''. If we choose $r = -a \omega$, then the velocity is simply:

$$v(t) = \frac{d h(t)}{d t} = r \sin ( \omega t + \phi ) .$$ (4)

The velocity has two independent Fourier components [Walker, 1988] with amplitudes x and y:

$$v(t) = x \sin ( \omega t) + y \cos ( \omega t) .$$ (5)

From the trigonometric identity $\sin(A+B) = \sin A \cos B + \cos A \sin B$and equations (4) and (5) we immediately find that $x = r \; \cos \phi$ and $y = r \; \sin \phi$. Fig. 3 represents these quantities graphically. On this graph, the point (x, y) completely defines the state of the oscillator at any time t. It is important to keep in mind that x and yhave units of velocity.

  

Figure 3: Geometry for a simple harmonic oscillator.

A possible state of a harmonic oscillator is represented by point A. Its maximum velocity is r and its phase is $\phi$. This state may also be represented by the coordinate (x, y). Distances in this figure have units of velocity.

In this paper we use the Fourier components (x, y) rather than polar coordinates $(r, \phi)$ because the Fourier description is symmetrical (x and y have the same units of velocity) whereas polar coordinates are not (they have units of velocity and angle).

The energy of the oscillator can be found from the maximum velocity and the mass:

$$E = {\scriptstyle \frac{1}{2}}m v_{max}^2 .$$ (6)

[Feynman et al., 1963]. The total energy is also the sum of the energies of the two independent sinusoidal components in equation (5) [Stremler, 1982], and since according to equation (4) v_max = r,

$$E = {\scriptstyle \frac{1}{2}}m r^2 = {\scriptstyle \frac{1}{2}}m x^2 + {\scriptstyle \frac{1}{2}}m y^2 ,$$ (7)

so

 

r² = x² + y² . (8)

This equation shows that in a vacuum, where the total energy E is constant, the radius r is constant and the locus of the point (x, y) is a circle. That is, at a given energy the set of all possible phase angles $\phi$describes a circle of radius $r = \sqrt{\frac{2 E}{m}}$in a two dimensional velocity space whose axes are the amplitudes of the two independent Fourier components of the oscillator.